On the topology of random complexes built over stationary point processes

TL;DR

This paper investigates the homology of random simplicial complexes built over stationary point processes, revealing how dependence among points affects topological features compared to independent or Poisson cases.

Contribution

It extends previous results to general stationary point processes, analyzing how dependence influences the homology and Betti numbers of random complexes.

Findings

Dependence among points alters Betti number growth.

Results differ significantly from i.i.d. and Poisson models.

Implications for robustness in topological data analysis.

Abstract

There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a random point process in $\mathbb {R}^d$, and the edges and faces are determined according to some deterministic rule, typically leading to \v{C}ech and Vietoris-Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either i.i.d. observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction or repulsion, we find phenomena quantitatively different from those observed in the i.i.d. and Poisson cases. From the point of view of topological data analysis, our results seriously impact considerations of model (non)robustness for statistical inference. Our proofs rely on analysis of subgraph and component counts of stationary point processes, which are of independent interest in stochastic geometry.